374 Mr. 0. Heaviside on Electromagnetic Waves, and the 



when the frequency is great enough, and this tends to /^vj, 

 [L x heing the inductivity and v x the speed in the conductor, 

 whatever g and k may be, provided they are finite. Thus, 

 finally, 



0=/^ + ^ (280) 



represents the impedance, or ratio of fv to H a , which are now 

 in the same phase. 



At any distance outside we know the result by the dielec- 

 tric solution for an outward wave. But there is only super- 

 ficial disturbance in the conducting sphere. 



43. Resistance at the front of a wave sent along a wire. — In 

 its entirety this question is one of considerable difficulty, for 

 two reasons, if not three. First, although we may, for 

 practical purposes, when we send a wave along a telegraph- 

 circuit, regard it as a plane wave, in the dielectric, on account 

 of ih.Q great length of even the short waves of telephony, and 

 the great speed, causing the lateral distribution (out from the 

 circuit) of the electric and magnetic fields to be, to a great 

 distance, almost rigidly connected with the current in the 

 wires and the charges upon them ; yet this method of re- 

 presentation must to some extent fail at the very front of 

 the wave. Secondly, we have the fact that the penetra- 

 tion of the electromagnetic field into the wires is not 

 instantaneous ; this becomes of importance at the front of 

 the wave, even in the case of a thin wire, on account of the 

 great speed with which it travels over the wire *. The resist- 

 ance per unit length must vary rapidly at the front, being 

 much greater there than in the body of the wave ; thus 

 causing a throwing back, equivalent to electrostatic or "jar " 

 retardation. 



Now, according to the electromagnetic theory, the resist- 

 ance must be infinitely great at the front. Thus, alternate 

 the current sufficiently slowly, and the resistance is practically 

 the steady resistance. Do it more rapidly, and produce 

 appreciable departure from uniformity of distribution of 

 current in the wire, and we increase the resistance to an 

 amount calculable by a rather complex formula. But do it 



* The distance within which, reckoned from the front of the wave 

 backward, there is material increased resistance, we may get a rough idea 

 of by the distance travelled by the wave in the time reckoned to bring 

 the current-density at the axis of the wire to, say, nine tenths of the final 

 value. It has all sorts of values. It may be 1 or 1000 kilometres, 

 according to the size of wire and material. At the front, on the assump- 

 tion of constant resistance, the attenuation is according to e-R*/ 2 L ? R being 

 the resistance, and L the inductance of the circuit per unit length. Hence 

 the importance of the increased resistance in the present question. 



